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4   THE VAGUE TOPOLOGY       109

(ii) The vague closure of H in MC(X) is a compact metrizable space with
respect to the vague topology.

(i) This is an immediate consequence of the Banach-Steinhaus theorem

(ii) Let (Uw) be a sequence of relatively compact open sets in X which
cover X and are such that " c UB + 1 (3.18.3). Since every compact subset
K of X is contained in some Urt (3.16), each space Jf (X; K) can be identified
with a closed subspace of one of the Banach spaces #(On), hence with a closed
subspace of Jf (X; On). Now we know (7.4.4) that *((}) is separable, hence
the same is true of jf (X; On) (3.10.9). Let (fmn)m^ be a dense sequence in
^(X; Uw). To show that H is metrizable it is enough to show that the vague
topology on H can be defined by the pseudo-distances |</mn, /i  v>| (12.4.6).
This means that if gi (1 ^ i ^ p) is a finite sequence of functions belonging to
;fc(X), if jU0 is an element of H and r a real number >0, there exists a
Unite number of functions fmknk (l^k^q) such that the relations /*6 H
ud \<fmknk,H-o>\^$r(l ^k^q) imply |<^, ji - ^0>| g r (1 //>).
But the #f all belong to Jf (X; OM) for some fixed n, and the set of restrictions
}f the jueH to Jf(X;(),,) is equicontinuous, by (i) above and (
Hence the assertion follows from (12.15.7).

It remains to show (by the same reasoning as in (12.15.7)) that, when we
dentify H with its image L in the product space CNXN by means of the
napping /i^/mw, M> L is closed in CNXN. If (jik) is a sequence of points
>f H such that each of the sequences /mn, ]"*>)* i is convergent, then it
bllows from (12.15.7) that, in each Jf(X;On), the restrictions of the fik
converge to a continuous linear form. Hence the sequence (/**) converges
vaguely to a measure on X.                                                                Q.E.D,

We shall see later (13.20) that condition (i) in (13.4.2) can also be written
n the form |ju|(K) <; CK for each measure ju e H.

Notice also that (13.4.2(ii)) implies (13.4.1) as a particular case.
In particular:

13.4.3)   Let v be a positive measure on X. Then the set of complex measures
i such that \IJL\ f$ v is metrizable and compact with respect to the vague topology.

For this set is evidently bounded and closed in MC(X) in the vague

It should be noted that a bounded set of measures does not necessarily
atisfy the hypothesis of (13.4.3), of X, there exists a real number CK > 0